Introduction

In, the “bimonster” (the wreathed square of the Fischer-Griess monster group) was studied in terms of its representation as a quotient of a certain infinite Coxeter group. Here we shall use the representation of this Coxeter group as a hyperbolic reflection group to investigate both the bimonster and its subgroup 3Fi24.

Throughout the paper, we shall use the notation of for group structures. In section 1, we give a simple axiomatic definition of a group G, and deduce that G is generated by 16 involutions that satisfy the Coxeter relations of Figure 1. This allows us to represent them in Section 2 by reflections in certain vectors of a hyperbolic space (that is, a space with a Lorentzian metric).

This notation makes it easy to perform calculations with these elements. In Section 2, we shall find some relations that must hold in G, but are not consequences of the Coxeter relations, and will use these to establish many identities in G, which we express in terms of alias groups.

Our section 4 contains a short proof of the 26 node theorem of.

The remainder of the paper is devoted to the subgroup Y552 of G, which we shall show has the structure 3Fi24. By way of introduction, Section 5 is used to show that the smaller group Y551 has structure, by completely enumerating its root vectors. In Section 6, we describe the root vectors for 3Fi24, and compute the corresponding alias groups.

A certain element ωis defined in Section 7, and shown to generate a normal subgroup of order 3 in Y552.

Abstract

Introduction

Recently the author published a paper which showed how progress had been made towards proving that Y555 (which we redefine below) is a presentation for the wreath square of the Fischer-Griess Monster (which we call the Bimonster) and outlined a possible method of completing the proof. Since then the proof has indeed been completed, but by a different method: results announced by A. Ivanov at the 1990 Durham Conference, proved by showing the simple connectedness of a certain simplicial complex, meant that a slight strengthening of the results of was sufficient to complete the proof. This was achieved during the conference, and it therefore seems appropriate to publish it here in the conference proceedings.

We also take the opportunity to present proofs of two other results needed for which no full published version currently exists.

Summary of

We start by recalling some of the notation, terminology and (without proof) results of. Note that the numbering of the theorems has been changed. References contain many other useful results about subgroups of Y555.

We recall that a Coxeter group is generated by involutions corresponding to the nodes of a (Coxeter) diagram. The product of two generators has order 2 or 3 according as the corresponding nodes are unjoined or joined by a single unlabelled edge. (Other product orders are possible and correspond to other types of join.)

INTRODUCTION

Let G be a simple, connected algebraic group over an algebraically closed field F of characteristic p ≥ 0. Let P = LQ be a parabolic subgroup of G, where Q is the unipotent radical of P and L is a Levi subgroup of P. Here L acts on Q via conjugation. This induces an L-action on consecutive subquotients of the lower central series of Q. Provided with a suitable F-vector space structure these quotients can be regarded as L-modules. They are called internal Chevalley modules for L.

There exists a unique parabolic subgroup P− of G such that P∪P− = L. We refer to P− as the opposite of P and Q− = Ru(P−) is called the opposite unipotent radical of P. The internal Chevalley modules that occur in Q− are dual to the ones in Q.

In this note we describe some results from regarding the structure of orbits for the action of L on these modules and give some information on the associated stabilizers for arbitrary characteristic.

(1.1) A motivation for this is a result of R. Richardson asserting that L has only finitely many orbits on each of its internal Chevalley modules. We show that there is a close connection between the L-orbits on Q−/(Q−)′ and (P, P)-cosets of G. For details and further information we refer to.

We say that p is a ‘very bad’ prime for G, if p occurs as a structure constant in Chevalley's commutator relations for G. In this situation there are degeneracies in these relations affecting the structure of orbits. We assume throughout these notes that p is not very bad for G.

This article serves as an exposition of the main result of. In we give the first computer free proof of the uniqueness of groups of type J4. In addition we supply simplified proofs of some properties of such groups, such as the structure of certain subgroups.

A group of type J4 is a finite group G possessing an involution z such that H = CG(z) satisfies F*(H) = Q is extraspecial of order 213, H/Q is isomorphic to ℝ3 extended by Aut(M22), and zG ∩ Q ≠ {z}. We prove:

Main TheoremUp to isomorphism there exists at most one group of type J4.

Janko was the first to consider groups of type J4 in, where he established various properties of such groups. For example Janko showed that each group G of type J4 is simple, he determined the order of G, and he described the normalizers of all subgroups of G of prime order. However Janko left open the question of whether there exist groups of type J4 and whether all groups of type J4 are isomorphic. Thus Janko is said to have discovered J4 and indeed J4 was the last of the 26 sporadic simple groups to be discovered.

Around 1980, Conway, Norton, Parker, and Thackray proved the existence and uniqueness of J4 using extensive machine computation. This work is discussed briefly in. Norton et al. construct J4 as a linear group in 112 dimensions over the field of order 2. While the notion of 2-local geometry did not exist at that time, this geometry plays an implicit role in.

Introduction

The combinatorial objects called buildings were first introduced (cf.) to provide a geometric approach to complex simple Lie groups – in particular the exceptional ones – and later on, more generally, to isotropic algebraic simple groups (cf. eg.). The buildings which do arise in that way are spherical, that is, have finite Weyl groups. This assertion has a partial converse, proved in: roughly speaking, there is a one-to-one correspondence between the (isomorphism classes of) buildings of irreducible spherical type and rank r ≥ 3 and the algebraic absolutely simple groups of relative rank r, where the notion of algebraic simple groups must be suitably extended so as to include, for instance, classical groups over division rings of infinite dimension over their centre. In order to have a similar statement in the rank 2 case, one must impose an extra condition, the Moufang condition, on the buildings under consideration.

The correspondence in question is established via the classification of buildings of irreducible, spherical types and rank ≥ 3. For non-spherical buildings, a full classification is known only for the affine types, in rank ≥ 4 (cf.): buildings of such a type have a “spherical building at infinity”, which is the essential tool for classification. A construction procedure for buildings of more general types given in shows that there cannot be any hope for a complete classification of buildings of arbitrary types. On the other hand, there is another wide class of groups, the Kac-Moody groups, which give rise to buildings, usually of non-spherical and non-affine types; these are “concrete” objects, which one also wishes to characterize geometrically.